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Excerpt from the Proceedings of the COMSOL Conference 2009 Milan
Impedance Spectroscopy and Cell Constant of the Electrodes for Deep
Brain Stimulation
Eduard Vinter 1, Sabine Petersen 1, Jan Gimsa 2, Ursula van Rienen 1
1
Institute of General Electrical Engineering, University of Rostock, Albert-Einstein-Str. 2, D-18059
Rostock, Germany
2
Institute of Biology, University of Rostock, Gertruden Strasse 11A, D-18057 Rostock, Germany
*Eduard Vinter: Albert-Einstein-Str. 2, D-18059 Rostock, Germany, [email protected]
Keywords: deep brain stimulation (DBS),
impedance
spectroscopy,
cell
constant,
equivalent circuit, FEM.
insulated in urethane whith four stimulation
electrodes at the end and is placed in the specific
part of the brain. The pulse generator, which is
implanted under the skin of the chest, sends
electrical impulses to the specific brain region.
The impulses run along the extensions from the
pulse generator to the lead in the brain.
For better understanding of the electrical
process of DBS more and more numerical
simulations are applied. The understanding of
DBS is a basis for improving the geometry and
the parameters of stimulated leads, which results
in better healing. In our work we compare the
two models of the electrodes for DBS (model
3387 and 3389, Medtronic®) in sodium chloride
solution. This solution accords to the
physiological fluid in the brain.
In this paper, we report about a simple model
within isotropic medium, the theory and the main
equations for our problem, the numerical
solution and finally we represent our results.
1. Introduction
2. Model
Deep brain stimulation (DBS) is an electrical
stimulation of neurons in specific parts of the
brain to treat movement disorders such as
Parkinson's disease (PD), tremor and dystonia
[1]. PD is a neurodegenerative disease caused by
the destruction of nerve cells in a deep region of
the brain (Substantia Nigra) [1]. This destruction
results in synchronous firing of nerve cells in
other parts of the brain (Subthalamicus Nucleus
or Globus Pallidus). The primary symptoms of
PD are tremor, rigidity and bradykinesia. In the
beginning the illness is treated with
medicaments. During the course of the disease
medicaments loose their effecting and usually a
medical stimulator, a so called “brain
pacemaker“, is implanted in Subthalamicus
Nucleus or Globus Pallidus to reduce these
symptoms. The stimulation system consists of
three components: the lead, the generator, and an
extension. The lead is the coil shaped conductor
2.1 Problem type
Abstract: To achieve a deeper understanding of
the mechanism of the deep brain stimulation
(DBS) [1] scientists use more and more
numerical
simulations.
DBS
inhibits
overreaching brain activity via electric pulses
that send into the brain by electrodes [1].
Different electrode parameters such as geometry,
frequency of stimulated impulse or applied
voltage have a great influence on the size of the
stimulated volume. To compare electrodes of
different shape we use the cell constant [2] and
the impedance spectroscopy [2]. The latter,
describes properties of electrodes in frequency
range. For our first model we have chosen
sodium chloride solution as surrounding medium
in the simulations. This solution accords to the
physiological fluid in the brain.
Comparison of electrodes with different
geometries is provided using the impedance
spectroscopy or the cell constant [2]. Impedance
spectroscopy is an electrochemical method for
the analysis of the electrical properties of
electrodes and their surrounding medium. This
analysis provides quantitative information about
the conductance, the static properties of the
interfaces of a system, and its dynamic change
due to adsorption or transfer of charge. The
impedance of two charged electrodes in the
medium is determined from the equivalent
circuit shown in Fig. 1.
Electrostatics mode was solved within the
AC/DC (emes) Module:
∇ ⋅ ε r ε 0∇V = 0,
(3)
where ε 0 is the permittivity of vacuum, ε r is
the relative permittivity and V is the potential.
2.2 Geometry & Subdomain Parameters
Figure 1. Equivalent circuit for electrodes of deep
brain stimulation.
From the equivalent circuit the
impedance Z is calculated (Eq. (1)):
Z=
1
1
+ jωC
R
,
complex
(1)
where R is the resistivity of medium, ω is the
angular frequency, C is the capacity of
electrodes and j is the imaginary unit.
Neglecting the influence of the permittivity, the
resistivity is calculated from numerical
simulation of the stationary flow fields. In this
case the electric field is coupled to the electric
current by Ohm’s law [4], which results in
Poisson’s equation (2):
∇ ⋅ σ∇V = 0,
The simulation model consists of four stimulated
electrodes, the electrode’s shaft and a cylinder
with sodium chloride solution as surrounding
medium. The geometry of the simulated lead is
taken from the description of the Medtronic 3389
and 3387 electrodes (Medtronic® Inc,
Minneapolis, MN, USA) [3] and consists of four
metal contacts of platinum iridium each with a
diameter of 1.27 mm, a height of 1.5 mm and an
outer jacket tubing of urethane 80A (Fig. 2).
(2)
where σ is specific conductivity and V is
potential.
Poisson’s equation is solved with help of the
Conductive Media DC (emdc) application mode
of the AC/DC (emes) Module. Whereas the
voltage U between the electrodes is given, the
current I has to be determined numerically to
calculate the ohmic resistance R of the
equivalent circuit from Ohm’s law ( R = U / I ).
The current results from integration of current
density over the surface of the electrode (see also
section 2.5.2).
The capacity is calculated from simulation of
the electrostatic fields neglecting the influence of
the conductivity. For calculation of the
capacitance C of the equivalent circuit using the
ratio of charge q divided by the voltage U
( C = q / U ), Poisson’s equation (3) for the
Figure 2. The 2D-electrodes for deep brain
stimulation (drawn after Implant manual, Medtronic®).
For the calculation of resistivity and capacity
we apply numerical integration over the specific
surface at the electrode. Each of the electrodes is
surrounded by a cylinder-sized closed surface for
the integration that has roughly the dimension of
the electrode. The separation between the
contacts is 1.5 mm. 2D axisymmetric finite
element models of the two DBS electrodes and
its surrounding medium were created using
COMSOL Multiphysics® 3.5a RC1 (Fig. 3).
2.4 Mesh & Solver
Figure 3. The 2D-simulation model (sector).
In the model we choose a cylinder of
70 × 77 mm in size with sodium chloride
solution for the surrounding medium, according
to the physiological fluid in the brain. For the
calculation of the resistivity in the Conductive
Media DC application mode the surrounding
medium and each of the stimulating electrodes
were modeled as conductors with conductivities
of 0.005 S/m and 1e6 S/m respectively. The
outer jacket tubing was modeled as an insulator
with conductivity 1e-15 S/m. For the calculation
of the capacity C in the Electrostatics application
mode the relative permittivity is set to 80 for
sodium chloride solution, 1 for platinum and 5
for urethane. Instead of the inhomogeneous and
anisotropic materials of the brain the sodium
chloride solution in this simulation is
homogeneous and isotropic.
2.3 Boundary conditions
For both application modes the boundary
conditions for the electrodes are the same. The
typical
electrode’s
polarity
is
bipolar
(stimulation electrode “0” is negative,
stimulation electrode “3” is positive) and
unipolar (electrode “0” is negative, “case” is
positive). In our model, two neighbors (electrode
“0” and “1”, see Fig. 2) of four stimulation
electrodes were specified as electric potential of
±1 V. Because the two remaining electrodes are
floating, they are chosen as continuity. It means
that they were not assigned a potential and were
ungrounded. The electrode shaft was defined as
continuity, too. We assume that our simulation
takes place in a glass cylinder within sodium
chloride solution. The walls of the glass were
assumed as zero charge in the Electrostatics
mode and as electric insulation in the Conductive
Media mode at the boundary.
The model was discretised with the triangle
method. The number of mesh elements consists
of 57295 elements. The cylinder was discretised
with a maximal element size of 4e-4. Because of
electric field values vary strongly on the
crossover from electrode to sodium chloride
solution, the stimulated electrodes and surface
for integration have to discretised relatively fine.
Maximum element size for each of the
stimulated electrodes and surfaces of integration
is 5e-5. To avoid numerical problems the
integration does not take place on the surface of
the electrode. Instead of that the surface of
integration is positioned at the distance of 0.1
mm to the electrode surface. The main equations
(see section 2.1) were solved using the iterative
solver GMRES with algebraic multigrid
preconditioner. The standard accuracy limit of
1e-6 had to be changed to 1e-8 in order to
achieve convergence. The surface of integration
was positioned at a distance of 0.1 mm from the
electrode and the area in-between was discretised
with finer mesh elements.
2.5 Postprosessing
2.5.1 Cell constant
One convenient characterization of the electrodes
is an electrode geometry factor or so called cell
constant γ [2]. The usability of the cell constant
is easily described by the example of a plate
capacitor (Fig. 4).
Figure 4. The plate capacitor.
If we have a chamber with two coplanar
electrodes and medium with conductivity σ
between the electrodes then according to Ohm’s
law the resistance R is determined using
equation (4):
R=
U
l
1
=
=
,
I σA σγ
(4)
where U is the applied voltage, I is the current
through the medium, l is the distance between
the electrodes, A is the area of electrodes, σ
the specific conductivity of the medium and γ
the cell constant. Rearranging equation (4)
results in the cell constant:
γ =
A
.
l
(5)
Other applications lead to other dependency of
the geometry. Equation (5) is valid for the plate
capacitor. The more the actual electrode
geometry differs from the plate capacitor the less
equation (5) may be used. Therefore in general,
the cell constant is computed from equation (6):
γ=
1
σR
2.5.2 Impedance spectroscopy
The impedance of the electrodes is usually as
imaginary vs. real part of the impedance. As
mentioned above, the electrode impedance can
be determined from the equivalent circuit (shown
in Fig. 1). The real part of the impedance is
represented by the term R (resistance) and the
imaginary part by the term jωC ( C capacity).
Term R represents the ionic bulk conductance
[2]. R and C are calculated from the current I
and the charge q for the given potential
difference U :
U
,
I
q
C= .
U
R=
(7)
(8)
The current I and the charge q were
computed using the area integral over the current
density J and the area integral over the dielectric
displacement D :
I=
∫∫ J ⋅ dA ,
(9)
∫∫ D ⋅ dA ,
(10)
A
,
(6)
q=
A
were R is taken from a numerical field
computation.
An advantage of the cell constant is the
independency from material parameters (for
isotropic medium) and reciprocal proportionality
to the resistance. This allows calculating the cell
constant from either a computed or a measured
resistance and the conductivity of the medium.
For DBS electrodes, the cell constant was
determined numerically from the current from
one electrode to another and the given potential
difference. The current was obtained by
integration over the surface of integration (see
section 2.2).
We expect that the cell constant of model
3387 is bigger than that of model 3389. The
reason for that is that model 3387 has a bigger
distance between the stimulation contacts than
model 3389 (see Eq. (5)).
where A is the area of integration.
3. Results and Discussion
3.1 Impedance
In this part we present our results for models
3387 and 3389 of the electrodes used for DBS.
Two of four electrodes are charged. Fig. 5 shows
imaginary vs. real part of the impedance in the
frequency range from 100 Hz to 5 MHz.
electric field distribution and the size of
stimulated volume.
3.1 Cell constant
Figure 5. Imaginary vs. real part of the impedance of
electrodes used for DBS (model 3387 and 3389,
Medtronic®).
The real part of the impedance dominates at
low frequencies. The resistance, the capacity and
the angular frequency at each point of the curve
are related by the ratio ω = 1 / RC . Fig. 6 shows
the real part of the impedance vs. frequency.
The calculated cell constants for model 3389 and
3387 are 0.0075 and 0.0061, respectively. As
could be expected, the cell constant is bigger for
model 3389 than for model 3387 which has a
larger distance between the stimulation contacts.
This means that model 3389 leads to a smaller
stimulated volume than model 3387 (for details
see section 2.5.1).
4. Conclusions
In this work a simple model for comparison
of different geometries of electrodes for DBS
was presented using parameters such as cell
constant and impedance.
Modeling of electrode’s effects for DBS is a
rather complex task. For a better description of
the stimulation effects, not only inhomogeneity
and anisotropy, but also the complex electric
properties of the brain tissue and electrochemical
processes have to be taken into account. For
example in the equivalent circuit the polarization
resistance and the double-layer capacitance were
not yet considered.
8. References
Figure 6. Real part of the impedance vs. frequency of
electrodes used for DBS (model 3387 and 3389,
Medtronic®).
The modeling and simulation of the
impedance spectroscopy with help of an
equivalent circuit affords information about the
electric properties of media like the electrolyte
and the electrochemical processes like charge
transfer. The different electrode shapes or
different distances between the electrodes cause
that the same electrode surface can conduct
current of different density. This has an impact
on the spread of the electric field and its
magnitude. This effect can then be considered
using cell constant [2]. Evidently, the cell
constant is one significant parameter for the
stimulation current. The same impedance at
reduced surface can be achieved by a decrease of
the electrode’s distance leading to the same
coefficient. This is directly correlated with
1. Joachim K. Krauss, Jens Volkmann, Tiefe
Hirnstimulation, 17-90. Steinkopff Verlag,
Darmstadt (2004)
2. Jan Gimsa, Beate Habel, Ute Schreiber, Ursula
van Rienen, Ulf Strauss and Ulrike Gimsa,
Choosing electrodes for deep brain stimulation
experiments – electrochemical considerations,
Journal of Neuroscience Methods, 142:2, 251265 (2005)
3. Implant manual, Medtronic Inc., 710
Medtronic
Parkway,
MN
55432-5604,
Minneapolis, USA, http://www.medtronic.com
4. Thorsten Steinmetz, Stefan Kurz, Markus
Clemens, Domains of Validity of Quasistatic and
Quasistationary Field Approximation, ISTET’09,
271-275 (2009)
9. Acknowledgements
This project is funded by the DFG in the
Research Training Group 1505/1.